arXiv subjects
Publications and source records attributed to R B Paris.
We consider some closed-form evaluations of certain infinite sums involving the Hurwitz zeta function $ζ(s,α)$ of the form \[\sum_{k=1}^\infty (\pm 1)^k k^m ζ(s,k),\] where $m$ is a non-negative integer. For the sums with $m=0$ and the argument $k$ in $ζ(s,k)$ replaced by $ka+b$, where $a$ and $b$ are positive parameters, we also obtain a transformation formula suitable for computation in the limit $a\to0$.
We investigate the asymptotic expansion of integrals analogous to Ball's integral \[\int_0^\infty \left(\frac{Γ(1+ν)|J_ν(x)|}{(x/2)^ν}\right)^{\!n}dx\] for large $n$ in which the Bessel function $J_ν(x)$ is replaced by the modified Bessel functions $I_ν(x)$ and $K_ν(x)$ together with appropriate exponential factors $e^{\mp x}$, respectively. The above integral with $J_ν(x)$ replaced by a hyper-Bessel function of the type recently discussed in Aktas {\it et al.} [The Ramanujan J., 2019] and taken over a finite interval determined by the first positive zero of the function is also considered for $n\to\infty$. We give the leading asymptotic behaviour of the hyper-Bessel function for $x\to+\infty$ in an appendix. Numerical examples are given to illustrate the accuracy of the various expansions obtained.
We consider the asymptotic expansion of the functional series \[S_{μ,γ}(a;λ)=\sum_{n=1}^\infty \frac{n^γe^{-λn^2/a^2}}{(n^2+a^2)^μ}\] for real values of the parameters $γ$, $λ>0$ and $μ\geq0$ as $|a|\to \infty$ in the sector $|\arg\,a|<π/4$. For general values of $γ$ the expansion is of algebraic type with terms involving the Riemann zeta function and a terminating confluent hypergeometric function. Of principal interest in this study is the case corresponding to even integer values of $γ$, where the algebraic-type expansion consists of a finite number of terms together with a contribution comprising an infinite sequence of increasingly subdominant exponentially small expansions. This situation is analogous to the well-known Poisson-Jacobi formula corresponding to the case $μ=γ=0$. Numerical examples are provided to illustrate the accuracy of these expansions.
We demonstrate how the asymptotics for large $|z|$ of the generalised Bessel function \[{}_0Ψ_1(z)=\sum_{n=0}^\infty\frac{z^n}{Γ(an+b) n!},\] where $a>-1$ and $b$ is any number (real or complex), may be obtained by exploiting the well-established asymptotic theory of the generalised Wright function ${}_pΨ_q(z)$. A summary of this theory is given and an algorithm for determining the coefficients in the associated exponential expansions is discussed in an appendix. We pay particular attention to the case $a=-1/2$, where the expansion for $z\to\pm\infty$ consists of an exponentially small contribution that undergoes a Stokes phenomenon. We also examine the different nature of the asymptotic expansions as a function of $\arg\,z$ when $-1<a<0$, taking into account the Stokes phenomenon that occurs on the rays $\arg\,z=0$ and $\arg\,z=\pmπ(1+a)$ for the associated function ${}_1Ψ_0(z)$. These regions are more precise than those given by Wright in his 1940 paper. Numerical computations are carried to verify several of the expansions developed in the paper.
We examine convergent representations for the sum of a decaying exponential and a Bessel function in the form \[\sum_{n=1}^\infty \frac{e^{-an}}{(\frac{1}{2} bn)^ν}\,J_ν(bn),\] where $J_ν(x)$ is the Bessel function of the first kind of order $ν>-1/2$ and $a$, $b$ are positive parameters. By means of a double Mellin-Barnes integral representation we obtain a convergent asymptotic expansion that enables the evaluation of this sum in the limit $a\to 0$ with $b<2π$ fixed. A similar result is derived for the sum when the Bessel function is replaced by the modified Bessel function $K_ν(x)$. The alternating versions of these sums are also mentioned.
We consider the asymptotic expansion of the generalised exponential integral involving the Mittag-Leffler function introduced recently by Mainardi and Masina [{\it Fract. Calc. Appl. Anal.} {\bf 21} (2018) 1156--1169]. We extend the definition of this function using the two-parameter Mittag-Leffler function. The expansions of the similarly extended sine and cosine integrals are also discussed. Numerical examples are presented to illustrate the accuracy of each type of expansion obtained.
We evaluate the sum of Gauss hypergeometric functions \[S(μ,c;x)=\sum_{k\geq 0} \bl(\frac{1-x}{1+μ}\br)^k\,{}_2F_1(\fs k+\fs, \fs k+1;c;x)\] for $x\in [-1,1]$ and positive parameters $μ$ and $c$. The domain of absolute convergence of this series is established by determining the growth of the hypergeometric function for $k\to+\infty$. An application to Galton-Watson branching processes arising in the theory of stochastic processes is presented. A new class of positive integer-valued distributions with power tails is introduced.
The refined asymptotic expansion of the confluent hypergeometric function $M(a,b,z)$ on the Stokes line $\arg\,z=π$ given in {\it Appl. Math. Sci.} {\bf 7} (2013) 6601--6609 is employed to derive the correct exponentially small contribution to the asymptotic expansion for the even and odd solutions of a second-order differential equation related to Weber's equation. It is demonstrated that the standard asymptotics of the parabolic cylinder function $U(a,z)$ yield an incorrect exponentially small contribution to these solutions. Numerical results verifying the accuracy of the new expansions are given.
We consider the asymptotic expansion of the Mathieu-Bessel series \[S_ν(a,b)=\sum_{n=1}^\infty \frac{n^γJ_ν(nb/a)}{(n^2+a^2)^μ}, \qquad (μ, b>0,\ γ, ν\in {\bf R})\] as $a\to+\infty$ with the other parameters held fixed, where $J_ν(x)$ is the Bessel function of the first kind of order $ν$. A special case arises when $γ+ν$ is a positive even integer, where the expansion comprises finite algebraic terms together with an exponentially small expansion. Numerical examples are presented to illustrate the accuracy of the various expansions. The expansion of the alternating variant of $S_ν(a,b)$ is considered. The series when the $J_ν(x)$ function is replaced by the Bessel function $Y_ν(x)$ is briefly mentioned.
We determine the asymptotic behaviour of the $n$th derivatives of the Bessel functions $J_ν(a)$ and $K_ν(a)$, where $a$ is a fixed positive quantity, as $n\to\infty$. These results are applied to the asymptotic evaluation of two incomplete Laplace transforms of these Bessel functions on the interval $[0,a]$ as the transform variable $x\to+\infty$. Similar evaluation of the integrals involving the Bessel functions $Y_ν(t)$ and $I_ν(t)$ is briefly mentioned.
We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by \[S_{ν,p}(a)=\sum_{n\geq 1} (an^p/2)^{-ν} K_ν(an^p)\qquad (a>0,\ 0\leqν<1)\] as the parameter $a\to 0+$, where $p$ denotes an integer satisfying $p\geq 2$. This extends previous work for the cases $p=1$ (linear) and $p=2$ (quadratic). The expansion as $a\to0+$ consists of an infinite number of asymptotic sums involving the Riemann zeta function, which when optimally truncated lead to remainder terms that are exponentially small in the parameter $a$. The number of these exponentially small terms associated with each optimally truncated asymptotic sum is found to increase with $p$.
We evaluate some integrals over $[0,\infty)$ of quotients of powers of the hyperbolic functions $\sinh x$ and $\cosh x$ using a hypergeometric approach. Some of these results appear to be new but several verify the entries in the table of integrals of Gradshteyn and Rhyzik.
The Brent-McMillan algorithm is the fastest known procedure for the high-precision computation of Euler's constant $γ$ and is based on the modified Bessel functions $I_0(2x)$ and $K_0(2x)$. An error estimate for this algorithm relies on the optimally truncated asymptotic expansion for the product $I_0(2x) K_0(2x)$ when $x$ assumes large positive integer values. An asymptotic expansion for this optimal error term is derived by exploiting the techniques developed in hyperasymptotics, thereby enabling more precise information on the error term than recently obtained bounds and estimates.
This paper deals with the evaluation of some definite Euler-type integrals in terms of the Wright hypergeometric function. We obtain a theorem on the Wright hypergeometric function and then use this theorem to evaluate some definite integrals. Further, we derive some results as applications of these evaluations. Multi-variable cases of the derived results of this paper are also briefly discussed.
We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[F(a+ελ,m;c+λ;x),\qquad λ\to+\infty\] for $x<1$ and positive integer $m$ when the parameter $ε>1$ and the constants $a$ and $c$ are supposed finite. When $m=1$, we employ the standard procedure of the method of steepest descents modified to deal with the situation when a saddle point is near a simple pole. It is shown that it is possible to give a closed-form expression for the coefficients in the resulting uniform expansion. The expansion when $m\geq 2$ is obtained by means of a recurrence relation. Numerical results illustrating the accuracy of the resulting expansion are given.
We consider the asymptotic behaviour of the generalised hypergeometric function \[{}_3F_2\bl(\!\!\begin{array}{c} 1, (1+t)k/2, (1+t)k/2+1/2\\tk+1, k+1\end{array}\!\!; x\br),\qquad 0<x,t\leq 1\] as the parameter $k\to+\infty$. Numerical results illustrating the accuracy of the resulting expansion are given.
In a recent paper, Dixit {\it et al.\/} [Acta Arith. {\bf 177} (2017) 1--37] posed two open questions whether the integral \[{\hat J}_{k}(α)=\int_0^\infty\frac{xe^{-αx^2}}{e^{2πx}-1}\,{}_1F_1(-k,3/2;2αx^2)\,dx\] for $α>0$ could be evaluated in closed form when $k$ is a positive even and odd integer. We establish that ${\hat J}_{k}(α)$ can be expressed in terms of a Gauss hypergeometric function and a ratio of two gamma functions, together with a remainder expressed as an integral. An upper bound on the remainder term is obtained, which is shown to be exponentially small as $k$ becomes large when $a=O(1)$.
We examine convergent representations for the sum of Bessel functions \[\sum_{n=1}^\infty \frac{J_μ(na) J_ν(nb)}{n^α}\] for $μ$, $ν\geq0$ and positive values of $a$ and $b$. Such representations enable easy computation of the series in the limit $a, b\to0+$. Particular attention is given to logarithmic cases that occur both when $a=b$ and $a\neq b$ for certain values of $α$, $μ$ and $ν$. The series when the first Bessel function is replaced by the modified Bessel function $K_μ(na)$ is also investigated, as well as the series with two modified Bessel functions.